Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning

The purpose of this paper is to further the notion of defining as a mathematical activity by elaborating a framework that structures the role of defining in student progress from informal to more formal ways of reasoning. The framework is the result of a retrospective account of a significant learning experience that occurred in an undergraduate geometry course. The framework integrates the instructional design theory of Realistic Mathematics Education (RME) and distinctions between concept image and concept definition and offers other researchers and instructional designers a structured way to analyze or plan for the role of defining in students’ mathematical progress.     

The Translation Invariant Massive Nelson Model: II. The Continuous Spectrum below the Two-Boson Threshold

In this paper we continue the study of the energy-momentum spectrum of a class of translation invariant, linearly coupled, and massive Hamiltonians from non-relativistic quantum field theory. The class contains the Hamiltonians of E. Nelson (J Math Phys 5:1190–1197, 1964) and H. Fröhlich (Adv Phys 3:325–362, 1954). In Møller (Ann Henri Poincaré 6:1091–1135, 2005; Rev Math Phys 18:485–517, 2006) one of us previously investigated the structure of the ground state mass shell and the bottom of the continuous energy-momentum spectrum. Here we study the continuous energy-momentum spectrum itself up to the two-boson threshold, the threshold for energetic support of two-boson scattering states. We prove that non-threshold embedded mass shells have finite multiplicity and can accumulate only at thresholds. We furthermore establish the non-existence of singular continuous energy-momentum spectrum. Our results hold true for all values of the particle-field coupling strength but only below the two-boson threshold. The proof revolves around the construction of a certain relative velocity vector field used to construct a conjugate operator in the sense of Mourre.     

Uranium assay and trace element analysis of the fourth collaborative material exercise samples by the modified Davies-Gray method and the ICP-MS/OES techniques

Journal of Radioanalytical and Nuclear Chemistry - An international group of laboratories participating in CMX-4 subjected three samples to comparative nuclear forensic analysis using uranium assay...     

1-Ethoxyethylidene, a new group for the stepwise SPPS of aminooxyacetic acid containing peptides

For more than a decade, the oxime ether ligation has proven to be one of the most efficient technique for the preparation of various peptide conjugates. However, despite numerous reports, the preparation of aminooxy-containing peptides is still hampered by N-overacylation of the NH-O function either during its incorporation or through the peptide-chain elongation. This restricts the introduction of protected-NH-O function at the last acylation step and prevents the use of standard solid-phase peptide synthesis (SPPS) procedures for the preparation of more complex aminooxy-peptides. We have studied the coupling of modified Fmoc-lysine containing either N-Boc- or N,N'-bis-Boc-protected aminooxyacetic acids (Aoa) during the elongation of the peptide chain and found that none of them is adequate. To circumvent this limitation, we propose to protect the Aoa moiety with a 1-ethoxyethylidene group (Eei) to provide 2-(1-ethoxyethylideneaminooxy)acetic acid building block. We showed that the Eei group is fully compatible with standard SPPS conditions and safely allows the multiple incorporation of the aminooxy functionality into the growing peptide. Since Eei-protected Aoa remains as flexible as normal amino acids in peptide synthesis, it may become the rule for the straightforward preparation of aminooxy peptides.     

Synthesis and characterization of new 2,3-disubstituted thieno[3,4-b]pyrazines: tunable building blocks for low band gap conjugated materials

Synthetic methods have been developed for the preparation of new 2,3-dihalothieno[3,4-b]pyrazines, from which a variety of new 2,3-difunctionalized thieno[3,4-b]pyrazines have been produced as precursors to conjugated materials. Structural, electronic, and optical characterization of these new analogues illustrate the extent to which the electronic nature of the functional groups can be used to tune the electronic properties of the thieno[3,4-b]pyrazine unit.     

Ruthenium-catalyzed azide-alkyne cycloaddition: scope and mechanism

The catalytic activity of a series of ruthenium(II) complexes in azide-alkyne cycloadditions has been evaluated. The [Cp*RuCl] complexes, such as Cp*RuCl(PPh 3) 2, Cp*RuCl(COD), and Cp*RuCl(NBD), were among the most effective catalysts. In the presence of catalytic Cp*RuCl(PPh 3) 2 or Cp*RuCl(COD), primary and secondary azides react with a broad range of terminal alkynes containing a range of functionalities selectively producing 1,5-disubstituted 1,2,3-triazoles; tertiary azides were significantly less reactive. Both complexes also promote the cycloaddition reactions of organic azides with internal alkynes, providing access to fully-substituted 1,2,3-triazoles. The ruthenium-catalyzed azide-alkyne cycloaddition (RuAAC) appears to proceed via oxidative coupling of the azide and alkyne reactants to give a six-membered ruthenacycle intermediate, in which the first new carbon-nitrogen bond is formed between the more electronegative carbon of the alkyne and the terminal, electrophilic nitrogen of the azide. This step is followed by reductive elimination, which forms the triazole product. DFT calculations support this mechanistic proposal and indicate that the reductive elimination step is rate-determining.     

Expanding the number of phthalates monitored in house dust

Phthalates have been used as plasticisers for several decades in various industry and consumer products. A method was developed for the determination of 13 not commonly monitored phthalates in household dust. The method was based on solvent extraction using sonication, sample clean-up by solid phase extraction (SPE), and analysis using isotope dilution gas chromatography-tandem mass spectrometry (GC/MS/MS). The method was applied to the analysis of dust samples collected using two vacuum sampling techniques from 38 urban Canadian homes: a sample of fresh or ‘active’ dust (FD) collected by technicians and a composite sample taken from the household vacuum cleaner (HD). Spearman rank correlations between HD and FD samples were significant for six phthalates with median concentrations above their method detection limits (MDLs), suggesting that the HD samples provide comparable results with FD samples. Seven phthalates were detected and quantified in a Canada-wide set of 126 household dust samples, among which six phthalates were detected at frequencies higher than 87%, with median (range) concentrations of 1.9 (<0.42–240) (μg/g) for diisohexyl phthalate (DIHxP), 3.8 (<0.16–260) (μg/g) for di-n-heptyl phthalate (DHepP), 6.6 (<1.1–1170) (μg/g) for diisooctyl phthalate (DIOP), 1.1 (<0.12–390) (μg/g) for di-n-octyl phthalate (DOP), 6.3 (<0.16–430) (μg/g) for dinonyl phthalate (DNP), and 1.8 (<0.18–850) (μg/g) for di-n-decyl phthalate (DDP). High detection frequencies and widely scattered concentration levels of these phthalates in this preliminary set of 126 samples suggested a high variability in potential exposure to phthalates in Canadian homes. NIST SRM 2585 (organic contaminants in house dust) was also analysed; eight phthalates were detected, with concentrations ranging from 6.0 μg/g for DOP to 79 μg/g for DIHxP. The results from SRM 2585 may contribute to the certification of phthalate concentration values in this SRM.     

Ruthenium-catalyzed cycloaddition of aryl azides and alkynes

The formation of 1,5-disubstituted 1,2,3-triazoles from aryl azides and alkynes was readily accomplished using [Cp*RuCl]4 catalyst in dimethylformamide. It was also demonstrated that the reaction provided higher yields, cleaner product, and shorter reaction times when carried out under microwave irradiation.     

Simulation of flux during electro-membrane extraction based on the Nernst-Planck equation

The present work has for the first time described and verified a theoretical model of the analytical extraction process electro-membrane extraction (EME), where target analytes are extracted from an aqueous sample, through a thin layer of 2-nitrophenyl octylether immobilized as a supported liquid membrane (SLM) in the pores in the wall of a porous hollow fibre, and into an acceptor solution present inside the lumen of the hollow fibre by the application of an electrical potential difference. The mathematical model was based on the Nernst-Planck equation, and described the flux over the SLM. The model demonstrated that the magnitude of the electrical potential difference, the ion balance of the system, and the absolute temperature influenced the flux of analyte across the SLM. These conclusions were verified by experimental data with five basic drugs. The flux was strongly dependent of the potential difference over the SLM, and increased potential difference resulted in an increase in the flux. The ion balance, defined as the sum of ions in the donor solution divided by the sum of ions in the acceptor solution, was shown to influence the flux, and high ionic concentration in the acceptor solution relative to the sample solution was advantageous for high flux. Different temperatures also led to changes in the flux in the EME system.     

Extrapolation methods for approximating arc length and surface area

A well-known method of estimating the length of a parametric curve in is to sample some points from it and compute the length of the polygon passing through them. In this paper we show that for uniform sampling of regular smooth curves, Richardson extrapolation can be applied repeatedly giving a sequence of derivative-free length estimates of arbitrarily high orders of accuracy. A similar result is derived for the approximation of the area of parametric surfaces.